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(D) The equations of the circles are $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$. Since the line $y = mx - b \sqrt{1 + m^2}$ is tangent to the first

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$\frac{2b}{\sqrt{a^2 - 4b^2}}$

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(D) The equations of the circles are $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$. Since the line $y = mx - b \sqrt{1 + m^2}$ is tangent to the first circle,it is a standard form for a tangent to $x^2 + y^2 = b^2$. For the second circle $(x - a)^2 + y^2 = b^2$,the tangent line equation is $y = m(x - a) \pm b \sqrt{1 + m^2}$. Since the line is common,we equate the two forms: $mx - b \sqrt{1 + m^2} = mx - ma \pm b \sqrt{1 + m^2}$. Taking the positive sign for the common tangent,we get: $ma = 2b \sqrt{1 + m^2}$. Squaring both sides: $m^2 a^2 = 4b^2 (1 + m^2) = 4b^2 + 4b^2 m^2$. $m^2 (a^2 - 4b^2) = 4b^2$. $m^2 = \frac{4b^2}{a^2 - 4b^2}$. Taking the positive square root: $m = \frac{2b}{\sqrt{a^2 - 4b^2}}$.

Given that $a > 2b > 0$ and that the line $y = mx - b \sqrt{1 + m^2}$ is a common tangent to the circles $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$. Then the positive value of $m$ is

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